Coexistence and configuration mixing in ¹¹²Cd

Abstract

I have performed a two-state mixing analysis between so-called rotational intruder $0^{+}$, $2^{+}$ and yrast $0^{+}$, $2^{+}$ states in ¹¹²Cd, fitting E2 transition matrix elements to obtain mixing amplitudes and matrix elements connecting basis states. Mixing is found to be small for both $0^{+}$ and $2^{+}$ states. Of two possible solutions, one provides a $0^{+}$ mixing amplitude of 0.251– identical to the one derived years ago from 2n transfer data.

Introduction

The even isotopes of Cd were long thought to be classic examples of harmonic vibrators, but their structure is now known to be very much more complex. Several Cd nuclei exhibit deformed intruder states near the energy of the two-phonon triplet. These are usually described as two-proton excitations out of the core [1], [2]. They are seen to form a K = 0 rotational band, and to mix with the lower, predominantly vibrational, states. Several workers have analyzed these states and their mixing [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15].

Heyde and Wood [4] have summarized various aspects of coexistence in many nuclei. In Cd nuclei, Heyde, et al. [13] concluded that the second and third $0^{+}$ states contained approximately equal amounts of two-phonon and rotational intruder configurations. I used 2n transfer data to estimate the amount of mixing between rotational intruder $0^{+}$ states and normal ground states [12]. Kotwal [11] expanded that analysis to allow three-state mixing. They concluded that the second $0^{+}$ state in ^{112, 114}Cd contained more than 90% of the intruder configuration, but that it was the third $0^{+}$ state in ¹¹⁶Cd. This result contradicted earlier conclusions [13]. Kumpulainen, et al. [5] concluded that two sets of low-lying $0^{+}$ states with different characteristics cross between ¹¹⁴Cd and ¹¹⁶Cd, in agreement with Kotwal, et al.

Délèze, et al. [6] presented new data for ¹¹²Cd and performed IBA-2 configuration-mixing calculations for ^{110, 112, 114}Cd. They took the second $0^{+}$ state to be the intruder in all three nuclei. Heyde, et al. [14] performed calculations for ^{112, 114}Cd in a basis of anharmonic quadrupole vibrations and more deformed intruder particle-hole excitations. They concluded that the pure interpretation of a two-phonon quadrupole $0^{+}$ state and an intruder $0^{+}$ state was to be replaced by a “perfect (1/2)^1/2, (1/2)^1/2 linear combination.” Garrett, et al. [9] interpreted the low-lying levels of ¹¹²Cd in a model that assumed mixing of normal phonon states and intruder configurations.

A recent paper [16] has provided new information for ¹¹²Cd, whose structure is quite similar to that of ¹¹⁴Cd [17]. It is clear from (t,p) transfer data [11], [18] that it is the second $0^{+}$ state that is predominantly of rotational character. This view is supported by E2 strengths, which require the third $2^{+}$ state to be the rotational one. Table 1 lists the E2 strengths for $0\leftrightarrow 2$ transitions and the transition matrix elements derived from them.

Table 1 and Fig. 1 compare $0\leftrightarrow 2$ transition matrix elements in ¹¹²Cd and ¹¹⁴Cd [19]. The similarity is apparent. Other relevant quantities in the two nuclei are compared in Table 2. In the ratio ΔE(2 phonon)/E(2₁), the quantity ΔE(2 phonon) is the average of the absolute values of 0-2 and 2-4 energy splittings in the supposed two-phonon triplet. It is a measure of the anharmonicity and/or mixing.

Previous work indicates that $0^{+}$ and $2^{+}$ states in all three bands have mixed, but for $0^{+}$, mixing of the third state with the other two was shown to be small [11]. Here, I consider the mixing of rotational intruder states with the lowest $0^{+}$ and $2^{+}$ states. The four relevant transition matrix elements are listed in Table 3. Some sign ambiguities can arise when taking square roots of B(E2) to get M(E2). In my phase convention, M₀ and M₃ are positive, whereas M₁ and M₂ can have either sign, because they involve destructive interference.